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Theorem cmbri 23084
Description: Binary relation expressing the commutes relation. Definition of commutes in [Kalmbach] p. 20. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)
Hypotheses
Ref Expression
pjoml2.1  |-  A  e. 
CH
pjoml2.2  |-  B  e. 
CH
Assertion
Ref Expression
cmbri  |-  ( A  C_H  B  <->  A  =  ( ( A  i^i  B )  vH  ( A  i^i  ( _|_ `  B
) ) ) )

Proof of Theorem cmbri
StepHypRef Expression
1 pjoml2.1 . 2  |-  A  e. 
CH
2 pjoml2.2 . 2  |-  B  e. 
CH
3 cmbr 23078 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_H  B  <->  A  =  ( ( A  i^i  B )  vH  ( A  i^i  ( _|_ `  B ) ) ) ) )
41, 2, 3mp2an 654 1  |-  ( A  C_H  B  <->  A  =  ( ( A  i^i  B )  vH  ( A  i^i  ( _|_ `  B
) ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652    e. wcel 1725    i^i cin 3311   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   CHcch 22424   _|_cort 22425    vH chj 22428    C_H ccm 22431
This theorem is referenced by:  cmcmlem  23085  cmcm2i  23087  cmbr2i  23090  cmbr3i  23094  pjclem1  23690  pjci  23695
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-iota 5410  df-fv 5454  df-ov 6076  df-cm 23077
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